2 PARTICULAR CASES AND EXAMPLES 253 We shall use this local definition of a Haar measure in Chapter XIX to construct a left Haar measure on a Lie group. Here we note the following consequence of (14.2.4): (14.2.5) Let G be a locally compact group, H a discrete normal subgroup of G, and n : G -* G/H the canonical homomorphism. Also let V be an open neighborhood of the neutral element of G such that the restriction of n to V is ahomeomorphism of V onto the neighborhood n(V) of the neutral element of G/H (12.11.2). Let A be a left Haar measure on G. If n is the image under n | V of the restriction lv of 1 to V, then ju is the restriction to n(V) of a left Haar measure on G/H. For every open set in n(V) is of the form 7r(U), where U c V is open, and the relation n(s)n(U) an(V) is equivalent to sU<=V; hence it follows immediately from the definitions that ju satisfies the condition of (14.2.4). Example (14.2.6) The mapping cp : t\-+e2Klt is a strict morphism (12.12.7) of R onto the compact group U of complex numbers of absolute value 1, by virtue of (9.5.2) and (9.5.7). The kernel of q> is the discrete subgroup Z consisting of the integers, and U may therefore be canonically identified with the quotient group R/Z = T (also called the \-dimensional torus or the additive group of real numbers modulo 1). Apply (14.2.5) to the case where V = ] —i, i[; bearing in mind that a Haar measure \JL on U must be diffuse (14.2.3) and that the complement of <p(V) in U consists of a single point, we see that a function /on U is ju-integrable if and only if the function t\-~*f(e2ltit) is Lebesgue- integrable on ]-i, if, and that we then have \fd\t. = f* 2 f(e2***) dt. J J —1/2 (14.2.7) Let G! , G2 be two locally compact groups, and ^ (resp. u2) a left Haar measure on Gj (resp. G2). Then /^ ®/^2 JS a left Haar measure on Gt x G2. For each function/e JTC(G! x G2) and each (sl9 s2) e Gt x G2 we have JI = dp^xi) f(slx^translation it follows that for all s, tin G the measures jus and ^